Interesting phenomenon

I dunno that removing the bulb would prove anything either way, unless you have a circuit diagram of the timer and suspect that the bulb load will affect the motor supply in some way.

I would suspect a simple-and-cheap-as-possible circuit, TBH, and it's prolly as straightforward as: Light on, fan supplied via connection through a switch, by-passing triac; light off, fan supplied via triac held on by timer circuit, which is prolly some extremely noddy one-shot comprising a reasonable sized capacitor discharging through a load (possibly the fan itself, or the triac, or something else very cheap).

At most, I'm guessing a 555 or similar. Either way, yer triac (or mebbe a relay with dodgy contacts) is prolly shorted by the light switch, making the motor go faster.